Problem: Find $\left(\sqrt{(\sqrt3)^3}\right)^4$.
Solution: Squaring the square root of any number gives that number back.  Therefore \[\left(\sqrt{(\sqrt3)^3}\right)^4=\left({\color{red}\left(\sqrt{{\color{black}(\sqrt3)^3}}\right)^2}\right)^2=\left((\sqrt3)^3\right)^2=(\sqrt3)^6.\] Again, squaring the square root gives the original number back so  \[(\sqrt3)^6=\left((\sqrt3)^2\right)^3=3^3=\boxed{27}.\]